Seminar

Counting partially directed walks in a symmetric wedge

Abstract

The enumeration of lattice paths in wedges poses unique mathematical challenges. These models are not translationally invariant, and the absence of this symmetry complicates both the derivation of a functional recurrence for the generating function, and its solution.

We consider a model of partially directed walks from the origin in the square lattice confined to a symmetric wedge defined by Y = ±X.

We derive a functional equation for the generating function of the model, and obtain an explicit solution using a version of the Kernel method.

This solution shows that there is a direct connection with matchings of an 2n-set counted with respect to the number of crossings, and a bijective proof has since been obtained.